Jost Solution and the Spectrum of the Discrete Dirac Systems

  • Elgiz Bairamov1Email author,

    Affiliated with

    • Yelda Aygar1 and

      Affiliated with

      • Murat Olgun1

        Affiliated with

        Boundary Value Problems20102010:306571

        DOI: 10.1155/2010/306571

        Received: 14 September 2010

        Accepted: 10 November 2010

        Published: 29 November 2010

        Abstract

        We find polynomial-type Jost solution of the self-adjoint discrete Dirac systems. Then we investigate analytical properties and asymptotic behaviour of the Jost solution. Using the Weyl compact perturbation theorem, we prove that discrete Dirac system has the continuous spectrum filling the segment [-2,2]. We also study the eigenvalues of the Dirac system. In particular, we prove that the Dirac system has a finite number of simple real eigenvalues.

        1. Introduction

        Let us consider the boundary value problem (BVP) generated by the Sturm-Liouville equation
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_Equ1_HTML.gif
        (1.1)
        and the boundary condition
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_Equ2_HTML.gif
        (1.2)
        where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq1_HTML.gif is a real-valued function and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq2_HTML.gif is a spectral parameter. The bounded solution of (1.1) satisfying the condition
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_Equ3_HTML.gif
        (1.3)
        will be denoted by http://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq3_HTML.gif . The solution http://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq4_HTML.gif satisfies the integral equation
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_Equ4_HTML.gif
        (1.4)
        It has been shown that, under the condition
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_Equ5_HTML.gif
        (1.5)
        the solution http://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq5_HTML.gif has the integral representation
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_Equ6_HTML.gif
        (1.6)
        where the function http://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq6_HTML.gif is defined by http://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq7_HTML.gif . The function http://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq8_HTML.gif is analytic with respect to http://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq9_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq10_HTML.gif , continuous http://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq11_HTML.gif , and
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_Equ7_HTML.gif
        (1.7)

        holds [1, chapter 3].

        The functions http://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq12_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq13_HTML.gif are called Jost solution and Jost function of the BVP (1.1) and (1.2), respectively. These functions play an important role in the solution of inverse problems of the quantum scattering theory [14]. In particular, the scattering date of the BVP (1.1) and (1.2) is defined in terms of Jost solution and Jost function. Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq14_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq15_HTML.gif , be the zeros of the Jost function, numbered in the order of increase of their moduli ( http://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq16_HTML.gif ) and
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_Equ8_HTML.gif
        (1.8)
        The functions
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_Equ9_HTML.gif
        (1.9)
        are bounded solutions of the BVP (1.1) and (1.2), where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq17_HTML.gif is the scattering function [14]. Using (1.7), we get that
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_Equ10_HTML.gif
        (1.10)

        hold. The collection of quantities http://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq18_HTML.gif that specify to as the behaviour of the radial wave functions http://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq19_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq20_HTML.gif at infinity is called the scattering of the BVP (1.1) and (1.2).

        Let us consider the self-adjoint system of differential equations of first order
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_Equ11_HTML.gif
        (1.11)

        where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq21_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq22_HTML.gif are real-valued continuous functions. In the case http://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq23_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq24_HTML.gif , where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq25_HTML.gif is a potential function and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq26_HTML.gif the mass of a particle, (1.11) is called stationary Dirac system in relativistic quantum theory [5, chapter 7]. Jost solution and the scattering theory of (1.11) have been investigated in [6].

        Jost solutions of quadratic pencil of Schrödinger, Klein-Gordon, and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq27_HTML.gif -Sturm-Liouville equations have been obtained in [79]. In [1017], using the analytical properties of Jost functions, the spectral analysis of differential and difference equations has been investigated.

        Discrete boundary value problems have been intensively studied in the last decade. The modelling of certain linear and nonlinear problems from economics, optimal control theory, and other areas of study has led to the rapid development of the theory of difference equations. Also the spectral analysis of the difference equations has been treated by various authors in connection with the classical moment problem (see the monographs of Agarwal [18], Agarwal and Wong [19], Kelley and Peterson [20], and the references therein). The spectral theory of the difference equations has also been applied to the solution of classes of nonlinear discrete Korteveg-de Vries equations and Toda lattices [21].

        Now let us consider the discrete Dirac system
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_Equ12_HTML.gif
        (1.12)
        with the boundary condition
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_Equ13_HTML.gif
        (1.13)
        where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq28_HTML.gif is the forward difference operator: http://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq29_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq30_HTML.gif is the backward difference operator: http://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq31_HTML.gif ; http://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq32_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq33_HTML.gif are real sequences. It is evident that (1.12) is the discrete analogy of (1.11). Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq34_HTML.gif denote the operator generated in the Hilbert space http://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq35_HTML.gif by the BVP (1.12) and (1.13). The operator http://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq36_HTML.gif is self-adjoint, that is, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq37_HTML.gif . In the following, we will assume that, the real sequences http://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq38_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq39_HTML.gif satisfy
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_Equ14_HTML.gif
        (1.14)

        In this paper, we find Jost solution of (1.12) and investigate analytical properties and asymptotic behaviour of the Jost solution. We also show that, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq40_HTML.gif , where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq41_HTML.gif denotes the continuous spectrum of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq42_HTML.gif , generated in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq43_HTML.gif by (1.12) and (1.13).

        We also prove that under the condition (1.14) the operator http://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq44_HTML.gif has a finite number of simple real eigenvalues.

        2. Jost Solution of (1.12)

        If http://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq45_HTML.gif for all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq46_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq47_HTML.gif from (1.12), we get
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_Equ15_HTML.gif
        (2.1)
        It is clear that
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_Equ16_HTML.gif
        (2.2)
        is a solution of (2.1). Now we find the solution http://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq48_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq49_HTML.gif of (1.12) for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq50_HTML.gif , satisfying the condition
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_Equ17_HTML.gif
        (2.3)

        where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq51_HTML.gif .

        Theorem 2.1.

        Under the condition (1.14) for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq52_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq53_HTML.gif ,  (1.12) has the solution http://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq54_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq55_HTML.gif , having the representation
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_Equ18_HTML.gif
        (2.4)
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_Equ19_HTML.gif
        (2.5)
        where
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_Equ20_HTML.gif
        (2.6)

        Proof.

        Substituting http://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq56_HTML.gif defined by (2.4) and (2.5) into (1.12) and taking http://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq57_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq58_HTML.gif , we get the following:
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_Equ21_HTML.gif
        (2.7)
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_Equ22_HTML.gif
        (2.8)
        Using (2.7) and (2.8),
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_Equ23_HTML.gif
        (2.9)
        hold, where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq59_HTML.gif . For http://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq60_HTML.gif , we obtain
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_Equ24_HTML.gif
        (2.10)

        By the condition (1.14), the series in the definition of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq61_HTML.gif ( http://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq62_HTML.gif ) are absolutely convergent. Therefore, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq63_HTML.gif ( http://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq64_HTML.gif ) can, by uniquely be defined by http://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq65_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq66_HTML.gif , that is, the system (1.12) for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq67_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq68_HTML.gif , has the solution http://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq69_HTML.gif given by (2.4) and (2.5).

        By induction, we easily obtain that
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_Equ25_HTML.gif
        (2.11)

        where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq70_HTML.gif is the integer part of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq71_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq72_HTML.gif is a constant. It follows from (2.4) and (2.11) that (2.3) holds.

        Theorem 2.2.

        The solution http://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq73_HTML.gif has an analytic continuation from http://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq74_HTML.gif to http://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq75_HTML.gif .

        Proof.

        From (1.14) and (2.11), we obtain that the series http://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq76_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq77_HTML.gif are uniformly convergent in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq78_HTML.gif . This shows that the solution http://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq79_HTML.gif has an analytic continuation from http://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq80_HTML.gif to http://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq81_HTML.gif .

        The functions http://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq82_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq83_HTML.gif are called Jost solution and Jost function of the BVP (1.12) and (1.13), respectively. It follows from Theorem 2.2 that Jost solution and Jost function are analytic in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq84_HTML.gif and continuous on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq85_HTML.gif .

        Theorem 2.3.

        The following asymptotics hold:
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_Equ26_HTML.gif
        (2.12)

        Proof.

        From (2.4), we get that
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_Equ27_HTML.gif
        (2.13)
        Using (2.11) and (2.13), we obtain
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_Equ28_HTML.gif
        (2.14)
        So we have
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_Equ29_HTML.gif
        (2.15)
        by (2.14). In a manner similar to (2.15), we get
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_Equ30_HTML.gif
        (2.16)

        From (2.15) and (2.16), we obtain (2.12).

        3. Continuous and Discrete Spectrum of the BVP (1.12) and (1.13)

        Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq86_HTML.gif denote the Hilbert space of all complex vector sequences
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_Equ31_HTML.gif
        (3.1)
        with the norm
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_Equ32_HTML.gif
        (3.2)

        Theorem 3.1.

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq87_HTML.gif .

        Proof.

        Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq88_HTML.gif denote the operator generated in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq89_HTML.gif by the BVP
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_Equ33_HTML.gif
        (3.3)
        We also define the operator http://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq90_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq91_HTML.gif by the following:
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_Equ34_HTML.gif
        (3.4)
        It is clear that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq92_HTML.gif and
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_Equ35_HTML.gif
        (3.5)
        where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq93_HTML.gif denotes the operator generated in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq94_HTML.gif by the BVP (1.12) and (1.13). It follows from (1.14) that the operator http://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq95_HTML.gif is compact in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq96_HTML.gif . We easily prove that
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_Equ36_HTML.gif
        (3.6)
        Using the Weyl theorem [22] of a compact perturbation, we obtain
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_Equ37_HTML.gif
        (3.7)
        Since the operator http://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq97_HTML.gif is selfadjoint, the eigenvalues of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq98_HTML.gif are real. From the definition of the eigenvalues, we get that
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_Equ38_HTML.gif
        (3.8)

        where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq99_HTML.gif denotes the set of all eigenvalues of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq100_HTML.gif .

        Definition 3.2.

        The multiplicity of a zero of the function http://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq101_HTML.gif is called the multiplicity of the corresponding eigenvalue of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq102_HTML.gif .

        Theorem 3.3.

        Under the condition (1.14), the operator http://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq103_HTML.gif has a finite number of simple real eigenvalues.

        Proof.

        To prove the theorem, we have to show that the function http://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq104_HTML.gif has a finite number of simple zeros.

        Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq105_HTML.gif be one of the zeros of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq106_HTML.gif . Now we show that
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_Equ39_HTML.gif
        (3.9)
        Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq107_HTML.gif be the Jost solution of (1.12) that is,
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_Equ40_HTML.gif
        (3.10)
        Differentiating (3.10) with respect to http://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq108_HTML.gif , we have
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_Equ41_HTML.gif
        (3.11)
        Using (3.10) and (3.11), we obtain
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_Equ42_HTML.gif
        (3.12)
        or
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_Equ43_HTML.gif
        (3.13)
        It follows from (3.13) that
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_Equ44_HTML.gif
        (3.14)

        that is, all zeros of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq109_HTML.gif are simple.

        Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq110_HTML.gif denote the infimum of distances between two neighboring zeros of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq111_HTML.gif . We show that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq112_HTML.gif . Otherwise, we can take a sequence of zeros http://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq113_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq114_HTML.gif of the function http://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq115_HTML.gif , such that
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_Equ45_HTML.gif
        (3.15)
        It follows from (2.4) that, for large http://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq116_HTML.gif ,
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_Equ46_HTML.gif
        (3.16)

        holds, where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq117_HTML.gif .

        From the equation
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_Equ47_HTML.gif
        (3.17)
        we get
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_Equ48_HTML.gif
        (3.18)

        There is a contradiction comparing (3.16) and (3.18). So http://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq118_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq119_HTML.gif function has only a finite number of zeros.

        Authors’ Affiliations

        (1)
        Department of Mathematics, Ankara University

        References

        1. Marchenko VA: Sturm-Liouville Operators and Applications, Operator Theory: Advances and Applications. Volume 22. Birkhäuser, Basel, Switzerland; 1986:xii+367.View Article
        2. Levitan BM: Inverse Sturm-Liouville Problems. VSP, Zeist, The Netherlands; 1987:x+240.
        3. Chadan K, Sabatier PC: Inverse Problems in Quantum Scattering Theory. Springer, New York, NY, USA; 1977:xxii+344.View Article
        4. Zaharov VE, Manakov SV, Novikov SP, Pitaevskiĭ LP: Theory of Solutions. Plenum Press, New York, NY, USA; 1984.
        5. Levitan BM, Sargsjan IS: Sturm-Liouville and Dirac Operators, Mathematics and Its Applications. Volume 59. Kluwer Academic Publishers, Dordrecht, The Netherlands; 1991:xii+350.View Article
        6. Gasymov MG, Levitan BM: Determination of the Dirac system from the scattering phase. Doklady Akademii Nauk SSSR 1966, 167: 1219-1222.MathSciNet
        7. Jaulent M, Jean C: The inverse s -wave scattering problem for a class of potentials depending on energy. Communications in Mathematical Physics 1972, 28: 177-220. 10.1007/BF01645775MathSciNetView Article
        8. Bairamov E, Karaman Ö: Spectral singularities of Klein-Gordon s -wave equations with an integral boundary condition. Acta Mathematica Hungarica 2002,97(1-2):121-131.MathSciNetView Article
        9. Adıvar M, Bohner M: Spectral analysis of q -difference equations with spectral singularities. Mathematical and Computer Modelling 2006,43(7-8):695-703. 10.1016/j.mcm.2005.04.014MathSciNetView Article
        10. Krall AM: A nonhomogeneous eigenfunction expansion. Transactions of the American Mathematical Society 1965, 117: 352-361.MathSciNetView Article
        11. Lyance VE: A differential operator with spectral singularities, I. AMS Translations 1967,2(60):185-225.
        12. Lyance VE: A differential operator with spectral singularities, II. AMS Translations 1967,2(60):227-283.
        13. Bairamov E, Çelebi AO: Spectrum and spectral expansion for the non-selfadjoint discrete Dirac operators. The Quarterly Journal of Mathematics. Oxford Second Series 1999,50(200):371-384. 10.1093/qjmath/50.200.371MathSciNetView Article
        14. Krall AM, Bairamov E, Çakar Ö: Spectrum and spectral singularities of a quadratic pencil of a Schrödinger operator with a general boundary condition. Journal of Differential Equations 1999,151(2):252-267. 10.1006/jdeq.1998.3519MathSciNetView Article
        15. Bairamov E, Çakar Ö, Krall AM: An eigenfunction expansion for a quadratic pencil of a Schrödinger operator with spectral singularities. Journal of Differential Equations 1999,151(2):268-289. 10.1006/jdeq.1998.3518MathSciNetView Article
        16. Bairamov E, Çakar Ö, Krall AM: Non-selfadjoint difference operators and Jacobi matrices with spectral singularities. Mathematische Nachrichten 2001, 229: 5-14. 10.1002/1522-2616(200109)229:1<5::AID-MANA5>3.0.CO;2-CMathSciNetView Article
        17. Krall AM, Bairamov E, Çakar Ö: Spectral analysis of non-selfadjoint discrete Schrödinger operators with spectral singularities. Mathematische Nachrichten 2001, 231: 89-104. 10.1002/1522-2616(200111)231:1<89::AID-MANA89>3.0.CO;2-YMathSciNetView Article
        18. Agarwal RP: Difference Equations and Inequalities: Theory, Methods, and Applications, Monographs and Textbooks in Pure and Applied Mathematics. Volume 228. 2nd edition. Marcel Dekker, New York, NY, USA; 2000:xvi+971.
        19. Agarwal RP, Wong PYJ: Advanced Topics in Difference Equations. Kluwer Academic Publishers, Dordrecht, The Netherlands; 1997.View Article
        20. Kelley WG, Peterson AC: Difference Equations: An Introduction with Applications. 2nd edition. Harcourt/Academic Press, San Diego, Calif, USA; 2001:x+403.
        21. Toda M: Theory of Nonlinear Lattices, Springer Series in Solid-State Sciences. Volume 20. Springer, Berlin, Germany; 1981:x+205.
        22. Glazman IM: Direct Methods of Qualitative Spectral Analysis of Singular Differential Operators. Israel Program for Scientific Translations, Jerusalem, Palestine; 1966:ix+234.

        Copyright

        © Elgiz Bairamov et al. 2010

        This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.